Set inversion and box contraction on Lie groups using interval analysis

Abstract

Set inversion and box contraction consist in finding guaranteed solutions to an equation where variables belong to bounded sets. These problems are classic in Euclidean spaces and can be solved using interval analysis. However, they quickly become complex when dealing with non-Euclidean manifolds. This paper investigates set inversion and box contraction problems when variables belong to generic Lie groups (e.g. the set of rotation matrices SO(3)). The concept of Lie group boxes is introduced and discussed. The core idea is to treat Lie group subsets via interval analysis in the Euclidean space of the Lie algebra. The concept of inclusion function on a Lie group is defined as an extension of classic inclusion functions in Euclidean interval analysis. It is shown that set inversion problems on Lie groups can be solved via the Lie algebra using Euclidean interval analysis and that the solution set has a finite volume in the group. On the same principle, the box contraction problem is extended to Lie groups via a Euclidean contractor in the Lie algebra. Numerical simulations demonstrate the high interest of using Lie group boxes instead of classic Euclidean parametrization for nonlinear problems (such as the Euler angles for rotations) both in terms of computational efficiency and accuracy. The proposed approach is of practical interest for state estimation and navigation in robotics since it makes it possible to solve highly nonlinear problems with a limited complexity and a high accuracy, in particular for attitude and pose estimation.